Application of the bootstrap method for solving simple quantum mechanical systems: a study of effectiveness

Abstract

This thesis evaluates the effectiveness of the quantum mechanical bootstrap by applying it to two fundamental systems with known analytical solutions: the simple harmonic oscillator and the hydrogen atom. The bootstrap method is a consistency-based approach for determining the energy spectra of quantum mechanical systems without explicitly solving the Schrödinger equation. The methodology involves deriving moment recursion relations from the Hamiltonian and canonical commutation relations, then constructing Hankel matrices from these moments. Physical energy eigenvalues are identified by imposing positivity constraints—requiring that these matrices be positive semi-definite, which ensures the moments correspond to a valid quantum state. Trial energies that violate this condition are systematically excluded, and the allowed regions converge toward the true spectrum as the matrix size increases. Our results demonstrate that the bootstrap method successfully reproduces the known energy spectra for both systems. The findings confirm that the bootstrap provides a reliable alternative approach for spectral determination in quantum mechanics, suggesting its potential applicability to more complex systems where analytical solutions are unavailable.